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...artition of G k can be obtained using various algorithms such as (a) spectral bisection [32, 31, 1, 18], (b) geometric bisection [28, 27] (if coordinates are available), and (c) combinatorial methods =-=[24, 2, 9, 10, 12, 4]-=-. Since the size of the coarser graph G k is small (i.e., |V k |s100), this step takes a small amount. METIS implements four different schemes for partitioning the coarsest graph, that are evaluated i...

... is not suitable for parallel direct methods, as it provides very little concurrency in the parallel factorization phase. METIS is a set of programs that implement the various algorithms described in =-=[22, 23]-=-. The advantages of METIS compared to other similar packages are the following: + Provides high quality partitions! The partitions produced by METIS are consistently 10% to 50% better than those produ...

... are used to solve a sparse system of equations, then a graph partitioning algorithm can be used to compute a fill reducing ordering that lead to high degree of concurrency in the factorization phase =-=[25, 10]-=-. The multiple minimum degree ordering used almost exclusively in serial direct methods is not suitable for parallel direct methods, as it provides very little concurrency in the parallel factorizatio...

...andom 33 33 33 21 SHEM Sorted Heavy-edge 33 33 33 51 SHEM* Sorted Modified Heavy-edge 3 33 33 Out of these schemes the first five of them are the main matching schemes that are described in detail in =-=[22, 23]-=-. The last three matching schemes are variations of RM, HEM and HEM*. These schemes before proceeding to find a matching using either RM, HEM, and HEM*, they first sort the vertices in non-decreasing ...

...se methods is the multiplication of a sparse matrix and a (dense) vector. Partitioning the graph corresponding to the matrix A, significantly reduces the amount of communication required by this step =-=[25]-=-. If parallel direct methods are used to solve a sparse system of equations, then a graph partitioning algorithm can be used to compute a fill reducing ordering that lead to high degree of concurrency...

...ions! The partitions produced by METIS are consistently 10% to 50% better than those produced by spectral partitioning algorithms [1, 19], and 5% to 15% better than those produced by Chaco multilevel =-=[20, 19]-=-. + It is extremely fast! In our experiments, on a wide range of graphs, we found that the multilevel recursive bisection algorithms # METIS is copyrighted by the regents of the University of Minnesot...

...A\A # # B # and B\B # # A # is a bisection with a smaller edge-cut. A class of algorithms that tend to produce very good results are those that are based on the Kernighan-Lin (KL) partition algorithm =-=[24, 7, 20]-=-. The KL algorithm is iterative in nature. It starts with an initial partition and in each iteration it finds subsets A # and B with the above properties. If such subsets exist, then it moves them to ...

...her similar packages are the following: + Provides high quality partitions! The partitions produced by METIS are consistently 10% to 50% better than those produced by spectral partitioning algorithms =-=[1, 19]-=-, and 5% to 15% better than those produced by Chaco multilevel [20, 19]. + It is extremely fast! In our experiments, on a wide range of graphs, we found that the multilevel recursive bisection algorit...

...gree of balance of the resulting elimination 8 tree to be displayed. The degree of balance is computed as an upper bound on the efficiency of a parallel algorithm that uses subtree-to-subcube mapping =-=[13]-=- (i.e., assuming communication cost to be zero). For instance, a balance factor of 0.654 on 64 processors, indicates that due to load imbalance alone, parallel Cholesky factorization algorithm on 64 p...

...artition of G k can be obtained using various algorithms such as (a) spectral bisection [32, 31, 1, 18], (b) geometric bisection [28, 27] (if coordinates are available), and (c) combinatorial methods =-=[24, 2, 9, 10, 12, 4]-=-. Since the size of the coarser graph G k is small (i.e., |V k |s100), this step takes a small amount. METIS implements four different schemes for partitioning the coarsest graph, that are evaluated i...

...res a lot of time, or does not produce very good results. Out of these matching schemes, HEM, HCM, and HEM* perform consistently better than either RM or LEM. In particular, the analysis presented in =-=[21]-=- shows that HEM does significantly better than RM, especially for 3D finite element graphs. On the other hand, the LEM matching scheme significantly increases the average degree of the coarser graphs,...

...plemented in METIS. Information about the other two schemes can be found in [22]. The first scheme, which is called random matching (RM), computes the maximal matching by using a randomized algorithm =-=[3, 20]-=-. The RM scheme works as follows. The vertices of the graph are visited in random order. If a vertex u has not been matched yet, then an unmatched adjacent vertex v is randomly selected and the edge (...

...artition of G k can be obtained using various algorithms such as (a) spectral bisection [32, 31, 1, 18], (b) geometric bisection [28, 27] (if coordinates are available), and (c) combinatorial methods =-=[24, 2, 9, 10, 12, 4]-=-. Since the size of the coarser graph G k is small (i.e., |V k |s100), this step takes a small amount. METIS implements four different schemes for partitioning the coarsest graph, that are evaluated i...

...her similar packages are the following: + Provides high quality partitions! The partitions produced by METIS are consistently 10% to 50% better than those produced by spectral partitioning algorithms =-=[1, 19]-=-, and 5% to 15% better than those produced by Chaco multilevel [20, 19]. + It is extremely fast! In our experiments, on a wide range of graphs, we found that the multilevel recursive bisection algorit...

...artition of G k can be obtained using various algorithms such as (a) spectral bisection [32, 31, 1, 18], (b) geometric bisection [28, 27] (if coordinates are available), and (c) combinatorial methods =-=[24, 2, 9, 10, 12, 4]-=-. Since the size of the coarser graph G k is small (i.e., |V k |s100), this step takes a small amount. METIS implements four different schemes for partitioning the coarsest graph, that are evaluated i...

...artition of G k can be obtained using various algorithms such as (a) spectral bisection [32, 31, 1, 18], (b) geometric bisection [28, 27] (if coordinates are available), and (c) combinatorial methods =-=[24, 2, 9, 10, 12, 4]-=-. Since the size of the coarser graph G k is small (i.e., |V k |s100), this step takes a small amount. METIS implements four different schemes for partitioning the coarsest graph, that are evaluated i...